Galerkin Method

The Galerkin method is a powerful numerical technique for approximating solutions to partial differential equations (PDEs) by projecting the problem onto a finite-dimensional subspace of test functions. Current research emphasizes hybrid approaches, combining Galerkin with finite element methods (FEM) and deep neural networks (DNNs) to improve accuracy and efficiency, particularly for time-dependent and nonlinear PDEs. These advancements are driving progress in diverse fields, including fluid dynamics, heat transfer, and neutron transport, by enabling faster and more accurate simulations of complex physical phenomena. Furthermore, research is focused on improving the method's applicability to high-dimensional problems and enhancing its convergence properties.